After the \(K^+\rightarrow \pi ^+\nu \bar{\nu }\)  selection, 471 events are
observed in the \(\pi ^+\pi ^0\)  region (Figure REF , left). The
kinematic factor is measured using a minimum-bias data sample with a
PNN-like selection applied. The \(\pi ^0\rightarrow \gamma \gamma \)  decay
is tagged by requiring exactly two photon clusters reconstructed in
the LKr, consistent with \(m_{\pi ^0}\)  assuming that the photons
originate from the \(K^+\)  decay vertex, and no other activity in SAC,
IRC or LAV. This provides a background-free sample of \(K^+\rightarrow \pi ^+\pi ^0\)  decays
selected with no constraints on the \(\pi ^+\)  kinematic variables. The
resulting \(m^2_{\textrm {miss}}\)  distribution is shown in
Figure REF , right. The assumption that the \(\pi ^0\)  tagging
is independent of the kinematics is tested by comparing the measured
kinematic factor in Region 1 with and without the \(\pi ^0\)  tagging
applied. The disagreement of 3\(\%\)  is assigned as a systematic
uncertainty. The background estimates in the signal and control
regions are obtained using equation REF  and are validated
by comparing expected and observed numbers of events within the
control regions. The presence of radiative \(K^+\rightarrow \pi ^+\pi ^0\gamma \)  decays from inner
bremsstrahlung increases the fraction of \(K^+\rightarrow \pi ^+\pi ^0\)  background. A correction
is applied to account for this effect using the simulated spectrum,
combined with a measurement of the energy-dependent single photon
detection efficiency of SAC, IRC, LAV and LKr on
data [1]}. The value of the correction represents 8\(\%\) 
of the \(K^+\rightarrow \pi ^+\pi ^0\)  background and a 100\(\%\)  systematic uncertainty is assigned
to account for the precision of the simulation.
